The mean of 100 observations was found to be 30. If two observations were wrongly taken as 32 and 12 instead of 23 and 11, find the correct mean.

Answer

To find the correct mean after correcting the wrongly taken observations, we can follow these steps:

### Step 1: Calculate the initial sum of observations
Given that the mean of 100 observations is 30, we can calculate the sum of observations using the formula:
\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} \]
Substituting the values:
\[ \text{Sum} = 30 \times 100 = 3000 \]

### Step 2: Identify the incorrect and correct observations
The two observations that were taken incorrectly are:
- Wrongly taken: 32 (instead of 23)
- Wrongly taken: 12 (instead of 11)

### Step 3: Calculate the sum with correct observations
To find the correct sum of observations, we will first add the correct observations (23 and 11) to the initial sum:
\[ \text{Correct sum} = \text{Initial sum} + \text{Correct observations} \]
\[ \text{Correct sum} = 3000 + 23 + 11 = 3000 + 34 = 3034 \]

### Step 4: Subtract the incorrect observations from the correct sum
Next, we need to subtract the incorrect observations (32 and 12) from the corrected sum:
\[ \text{Final sum} = \text{Correct sum} - \text{Incorrect observations} \]
\[ \text{Final sum} = 3034 - (32 + 12) = 3034 - 44 = 2990 \]

### Step 5: Calculate the correct mean
Now we can find the correct mean using the final sum and the number of observations:
\[ \text{Correct Mean} = \frac{\text{Final sum}}{\text{Number of observations}} \]
\[ \text{Correct Mean} = \frac{2990}{100} = 29.90 \]

### Final Result
The correct mean after adjusting for the errors is **29.90**.

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